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REVIEW 4 major objections 3 minor 6 cited by

Extended thermodynamical topology of black hole

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Three topological approaches to black hole thermodynamics are unified, and the zeros of a k-th order vector field encode the critical exponents.

desk verdict A plausible unification claim in black hole topological thermodynamics that I can't verify from what I was sent: the abstract promises a real quantitative correspondence, but the full text provided is a different paper, so the referee's job is to audit the zero-set regularity and the non-circularity of the critical-exponent match. read the letter →

arxiv 2508.01614 v1 pith:DSFE327C submitted 2025-08-03 hep-th gr-qc

Shan-Ping Wu , Si-Jiang Yang , Shao-Wen Wei This is my paper
classification hep-thgr-qc PACS 04.70.-s05.70.Fh04.50.-h
keywords blackholethermodynamicsextendedthermodynamicaltopologytopologicalinvariantsk-thordervectorfieldcriticalexponentsLovelockgravityphasetransitionswindingnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Thermodynamical topology classifies black hole phase behavior through topological invariants, but three separate invariants have been used for three different targets: the phases themselves, the spinodal curves, and the critical points. This paper argues that these three are facets of a single structure, which it calls extended thermodynamical topology, built from a k-th order vector field. On Einstein-gravity black holes the unified framework reproduces the phase structure in topological language, and on seven-dimensional Lovelock black holes it brings out novel phase phenomena. The central payoff is a correspondence between the zeros of that vector field and the thermodynamic critical exponents, so that critical behavior can be read off topologically. A sympathetic reader would care because this turns classification of black hole phase transitions into a single winding-number computation.

What carries the argument

The central object is the k-th order vector field on the thermodynamic parameter space, whose zeros are the topologically interesting points (phase-transition or critical points), with the winding number of the field around each zero serving as the topological invariant. Extending the construction to all orders k is what integrates the previously separate invariants: different k pick out different thermodynamic features, and the hierarchy of zero structures constitutes the extended thermodynamical topology. The correspondence between these zeros and critical exponents is the mechanism that connects the topological classification to quantitative critical behavior.

What would settle it

Take a known black hole family with a standard critical point, compute the winding number of the k-th order vector field around the critical point, and check whether the predicted divergence exponents match the known heat-capacity critical exponents; a mismatch, or a zero whose winding number changes continuously as parameters vary, would falsify the claimed correspondence. A sharper test: find an example where two zeros collide; if the unified invariant does not reproduce the known swallowtail phase structure at the collision, the integration of the three invariants fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that the three existing topological invariants of black hole thermodynamics—one for equilibrium phases, one for spinodal curves, one for critical points—are not independent classifications but special cases of one underlying extended thermodynamical topology. In this framework, a k-th order vector field is defined on thermodynamic parameter space, and its zeros carry winding numbers that label the phase structure; the order k selects the level of thermodynamic refinement. Applying the framework to black holes in Einstein gravity gives a systematic topological account of their phase structure, and applying it to black holes in seven-dimensional Lovelock gravity produces new thermodynamic phenomena that appear naturally from the topological data. The paper further establishes a correspondence between the zeros of the k-th order vector field and the critical exponents, meaning the fine structure of the vector field encodes how thermodynamic quantities diverge at criticality.

Load-bearing premise

The load-bearing premise is that the three topological invariants can be combined into one framework and that the k-th order vector field has isolated zeros with well-defined winding numbers across the whole thermodynamic parameter space; if zeros merge or become non-isolated, the unified classification and the correspondence with critical exponents would not hold.

Editorial extensions

If this is right

  • One winding-number computation in extended thermodynamical topology replaces separate topological analyses of phases, spinodal curves, and critical points.
  • Critical exponents can be associated with the zero structure of the k-th order vector field, giving a topological way to identify universality-class data.
  • The Einstein-gravity application provides a complete topological phase diagram for those black holes, with each phase transition appearing as a zero with a definite winding number.
  • The seven-dimensional Lovelock application shows that exotic phase behavior in higher-curvature gravity is not an artifact of analysis but persists as topologically required structure.
  • The framework supplies a physical interpretation for why thermodynamical topology works, rather than treating the invariants as formal labels.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the zero-to-exponent correspondence holds beyond the examples considered, critical exponents could be computed directly from winding data, which would bypass solving the equation of state in some regimes; this is an extension the paper does not fully develop.
  • The hierarchy of k-th order zeros resembles a refinement of thermodynamic response functions, so one could test the framework against geometric approaches such as thermodynamic curvature by comparing where each predicts critical behavior.
  • The framework invites a classification program: black holes sharing the same extended topological data would be expected to share phase-transition behavior, even across different gravity theories; this is testable in other Lovelock dimensions or Gauss-Bonnet black holes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The manuscript (arXiv:2508.01614) as represented by its abstract proposes a unified framework, termed "extended thermodynamical topology," that integrates three previously separate topological approaches to black hole thermodynamics: those classifying phases, spinodal curves, and critical points. The authors claim to apply this framework to black holes in Einstein gravity and in 7-dimensional Lovelock gravity, and further claim a correspondence between zeros of a k-th order vector field and thermodynamic critical exponents. The abstract presents this as a robust and fine-grained classification scheme. However, the "full text" supplied for review is a different manuscript entirely: it is the book manuscript of De Haro and Butterfield on the philosophy and physics of duality (arXiv:2508.01616), not the black hole topology paper. Consequently, no derivations, definitions, or numerical results of the claimed framework are available for inspection in the review package.

Significance. If the claims are correct, the paper could provide a useful conceptual unification of several active lines of work on black hole thermodynamic topology, and the proposed correspondence between vector-field zero structure and critical exponents would be a substantive new result, particularly if it holds beyond the two example theories. However, the significance cannot be assessed from the available material: the abstract alone contains no definitions, equations, or checks. The supplied full text is unrelated to the abstract, so the central claims are unverifiable in this review. The paper would need to demonstrate that the three topological invariants are indeed consistently integrated, that the k-th order vector field has well-defined winding numbers (e.g., isolated, non-degenerate zeros), and that the critical exponents are computed independently of the vector field to avoid circularity.

major comments (4)
  1. [Full Text (supplied)] The full text provided for review is not the manuscript under consideration: it is the book manuscript "The Philosophy and Physics of Duality" by De Haro and Butterfield (arXiv:2508.01616), as explicitly stated in its header. This is a load-bearing issue because none of the central claims of the abstract—the unified framework, the definition of the k-th order vector field, the topological classifications, or the correspondence with critical exponents—can be checked. The authors (or the editorial office) must supply the actual text of arXiv:2508.01614 before the paper can be evaluated.
  2. [Abstract] The abstract does not state any regularity condition for the zeros of the k-th order vector field. The skeptic's concern is concrete: for the winding number of a vector field to be well defined, its zeros must be isolated and non-degenerate (or handled by a consistent resolution) in the relevant parameter space. In higher-dimensional or Lovelock gravity, multiple critical points and swallowtail structures can cause the Jacobian of the k-th order vector field to vanish on curves, making zeros non-isolated. The manuscript needs to state and prove a genericity or regularity condition under which the zeros are isolated for the examples considered, and clarify whether the topological index remains meaningful at degenerate points.
  3. [Abstract] The claimed "correspondence between the zeros of the k-th order vector field and the associated critical exponents" risks circularity if the critical exponents are themselves extracted from the vector-field construction. The manuscript must define how critical exponents are obtained independently (e.g., from the scaling behavior of heat capacity or compressibility near the critical point) and then show that the vector-field zero structure reproduces them in a non-tautological way. Without such an independent definition, the correspondence could be an artifact of the construction.
  4. [Abstract] The abstract states that three distinct topological invariants are integrated, but it does not name them, specify their domains (e.g., parameter planes such as (P, v) or (T, S)), or define the k-th order vector field. A unified framework needs a precise statement of how the three invariants are related: whether they are derived from a single master vector field, or whether the unification is at the level of interpretation. The missing definitions make it impossible to assess the novelty or consistency of the claimed integration.
minor comments (3)
  1. [Abstract] The phrase "three distinct yet complementary topological invariants" is vague; the manuscript should cite the specific prior constructions (e.g., Duan's topological current method, the off-shell free energy approach, and the Hawking-Page/spinodal classification) to locate the contribution.
  2. [Abstract] The symbol k in "k-th order vector field" is not defined in the abstract; the manuscript should clarify what order means here (e.g., derivative order of the free energy) and specify the range of k considered.
  3. [Abstract] The word "thermodynamical" appears in the paper title and abstract; the more standard spelling in the modern literature is "thermodynamic," though this is a stylistic point.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity established: the supplied full text is a different paper, and the abstract alone provides no derivational chain to audit.

full rationale

The full text supplied is not the target manuscript. It is arXiv:2508.01616, the book manuscript 'The Philosophy and Physics of Duality' by De Haro and Butterfield, whereas the target is arXiv:2508.01614, 'Extended thermodynamical topology of black hole' by Wu, Yang, and Wei. The only in-scope evidence from the target paper is its abstract. The abstract asserts that three topological approaches are integrated and that a correspondence exists between zeros of the k-th order vector field and critical exponents, but it contains no equations, no definitions of the vector field or of the critical exponents, and no derivation showing that either object is defined in terms of the other. Without the actual manuscript, no specific reduction can be quoted, so the hard rule against non-quoted circularity speculation applies. The full-text mismatch is a limitation of the supplied material, not a circular step, and is flagged here as required by the reviewing rule. No fitted parameter is renamed as a prediction, no load-bearing self-citation is visible, and no uniqueness theorem is invoked. Therefore no significant circularity is established; the honest finding is a non-finding with score 0.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The abstract-only evidence shows no free parameters or invented physical entities. The single listed axiom is the validity and compatibility of the three prior topological invariants, which the paper treats as established input. The absence of derivations prevents identification of additional assumptions, fitted constants, or new entities.

assumptions (1)
  • domain assumption The three prior topological invariants (for phases, spinodal curves, and critical points) are valid and mutually consistent characterizations of black hole thermodynamics.
    The abstract builds on these established invariants and integrates them without re-deriving their validity.

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Cite this review

Pith. "Pith review of Extended thermodynamical topology of black hole." pith.science (2026). https://pith.science/paper/DSFE327C

@misc{pith2026250801614,
  author       = {Pith},
  title        = {Pith review of: Extended thermodynamical topology of black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DSFE327C}},
  note         = {Machine review of arXiv:2508.01614}
}
abstract

Thermodynamical topology has emerged as a powerful framework for classifying the thermodynamical behavior of black holes. Three distinct yet complementary topological invariants have been employed to characterize black hole phases, spinodal curves, and critical points in black hole thermodynamics. In this work, we develop a unified framework that integrates these three topological approaches and introduce the concept of extended thermodynamical topology, providing a clear physical interpretation. As a first step, we apply this framework to black holes in Einstein gravity, systematically elucidating their phase structure in terms of topological invariants. We then extend our analysis to black holes in 7-dimensional Lovelock gravity, where novel thermodynamic phenomena naturally emerge from the topological perspective. Moreover, we explore the connection between critical exponents and the extended thermodynamical topology, uncovering a correspondence between the zeros of the $k$-th order vector field and the associated critical exponents. Our study demonstrates that extended thermodynamical topology offers a robust and fine-grained framework for analyzing and classifying black hole phase transitions.

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Reference graph

Works this paper leans on

63 extracted references · 18 canonical work pages · cited by 5 Pith papers

  1. [1]

    J. D. Bekenstein, Black holes and entropy , Phys. Rev. D 7, 2333 (1973)

    work page 1973

  2. [2]

    S. W. Hawking, Particle Creation by Black Holes , Commun. Math. Phys. 43, 199 (1975), [erratum: Commun. Math. Phys. 46, 206 (1976)]

    work page 1975

  3. [3]

    N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space , Cambridge University Press, 1982,

    work page 1982

  4. [4]

    J. B. Hartle and S. W. Hawking, Path Integral Derivation of Black Hole Radiance , Phys. Rev. D 13, 2188 (1976)

    work page 1976

  5. [5]

    G. W. Gibbons and S. W. Hawking, Action Integrals and Partition Functions in Quantum Gravity , Phys. Rev. D 15, 2752 (1977)

    work page 1977

  6. [6]

    S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-De Sitter Space , Commun. Math. Phys. 87, 577 (1983)

    work page 1983

  7. [7]

    Chamblin, R

    A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Charged AdS black holes and catastrophic holography , Phys. Rev. D 60, 064018 (1999), [arXiv:hep-th/9902170 [hep-th]]

    arXiv 1999

  8. [8]

    Chamblin, R

    A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Holography, thermodynamics and fluctuations of charged AdS black holes , Phys. Rev. D 60, 104026 (1999), [arXiv:hep-th/9904197 [hep-th]]

    arXiv 1999

Show all 63 references
  1. [9]

    Kubiznak and R

    D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes , JHEP 07, 033 (2012), [arXiv:1205.0559 [hep-th]]

    2012 arXiv

  2. [10]

    Strominger and C

    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy , Phys. Lett. B 379, 99 (1996), [arXiv:hep-th/9601029 [hep-th]]

    1996 arXiv

  3. [11]

    Rovelli, Black hole entropy from loop quantum gravity , Phys

    C. Rovelli, Black hole entropy from loop quantum gravity , Phys. Rev. Lett. 77, 3288 (1996), [arXiv:gr-qc/9603063 [gr-qc]]

    1996 arXiv

  4. [12]

    't Hooft, A Planar Diagram Theory for Strong Interactions , Nucl

    G. 't Hooft, A Planar Diagram Theory for Strong Interactions , Nucl. Phys. B 72, 461 (1974)

    1974

  5. [13]

    't Hooft, Dimensional reduction in quantum gravity , Conf

    G. 't Hooft, Dimensional reduction in quantum gravity , Conf. Proc. C 930308, 284 (1993), [arXiv:gr-qc/9310026 [gr-qc]]

    1993 arXiv

  6. [14]

    Susskind, The World as a hologram , J

    L. Susskind, The World as a hologram , J. Math. Phys. 36, 6377 (1995), [arXiv:hep-th/9409089 [hep-th]]

    1995 arXiv

  7. [15]

    J. M. Maldacena, The Large N limit of superconformal field theories and supergravity , Adv. Theor. Math. Phys. 2, 231 (1998), [arXiv:hep-th/9711200 [hep-th]]

    1998 arXiv

  8. [16]

    Witten, Anti de Sitter space and holography , Adv

    E. Witten, Anti de Sitter space and holography , Adv. Theor. Math. Phys. 2, 253 (1998), [arXiv:hep-th/9802150 [hep-th]]

    1998 arXiv

  9. [17]

    Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories , Adv

    E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories , Adv. Theor. Math. Phys. 2, 505 (1998), [arXiv:hep-th/9803131 [hep-th]]

    1998 arXiv

  10. [18]

    Kubiznak, R

    D. Kubiznak, R. B. Mann, and M. Teo, Black hole chemistry: thermodynamics with Lambda , Class. Quant. Grav. 34, 063001 (2017), [arXiv:1608.06147 [hep-th]]

    2017 arXiv

  11. [19]

    M. R. Visser, Holographic thermodynamics requires a chemical potential for color , Phys. Rev. D 105, 106014 (2022), [arXiv:2101.04145 [hep-th]]

    2022 arXiv

  12. [20]

    W. Cong, D. Kubiznak, R. B. Mann, and M. R. Visser, Holographic CFT phase transitions and criticality for charged AdS black holes , JHEP 08, 174 (2022), [arXiv:2112.14848 [hep-th]]

    2022 arXiv

  13. [21]

    M. B. Ahmed, W. Cong, D. Kubiz n \'ak, R. B. Mann, and M. R. Visser, Holographic Dual of Extended Black Hole Thermodynamics , Phys. Rev. Lett. 130, 181401 (2023), [arXiv:2302.08163 [hep-th]]

    2023 arXiv

  14. [22]

    S. J. Yang, M. S. Ali, S. W. Wei, and Y. X. Liu, Holographic thermodynamics of a five-dimensional neutral Gauss Bonnet AdS black hole , Eur. Phys. J. C 85, 420 (2025), [arXiv:2408.00318 [hep-th]]

    2025 arXiv

  15. [23]

    D. G. Boulware and S. Deser, String Generated Gravity Models , Phys. Rev. Lett. 55, 2656 (1985)

    1985

  16. [24]

    J. T. Wheeler, Symmetric Solutions to the Maximally Gauss-Bonnet Extended Einstein Equations , Nucl. Phys. B 273, 732 (1986)

    1986

  17. [25]

    R. G. Cai, A Note on thermodynamics of black holes in Lovelock gravity , Phys. Lett. B 582, 237 (2004), [arXiv:hep-th/0311240 [hep-th]]

    2004 arXiv

  18. [26]

    R. C. Myers and B. Robinson, Black Holes in Quasi-topological Gravity , JHEP 08, 067 (2010), [arXiv:1003.5357 [gr-qc]]

    2010 arXiv

  19. [27]

    Oliva and S

    J. Oliva and S. Ray, A new cubic theory of gravity in five dimensions: Black hole, Birkhoff's theorem and C-function , Class. Quant. Grav. 27, 225002 (2010), [arXiv:1003.4773 [gr-qc]]

    2010 arXiv

  20. [28]

    R. A. Hennigar, D. Kubiz n \'ak, and R. B. Mann, Generalized quasitopological gravity , Phys. Rev. D 95, 104042 (2017), [arXiv:1703.01631 [hep-th]]

    2017 arXiv

  21. [29]

    Bueno, P

    P. Bueno, P. A. Cano, J. Moreno, and \'A. Murcia, All higher-curvature gravities as Generalized quasi-topological gravities , JHEP 11, 062 (2019), [arXiv:1906.00987 [hep-th]]

    2019 arXiv

  22. [30]

    Bueno, P

    P. Bueno, P. A. Cano, and R. A. Hennigar, (Generalized) quasi-topological gravities at all orders , Class. Quant. Grav. 37, 015002 (2020), [arXiv:1909.07983 [hep-th]]

    2020 arXiv

  23. [31]

    Bueno, P

    P. Bueno, P. A. Cano, R. A. Hennigar, M. Lu, and J. Moreno, Generalized quasi-topological gravities: the whole shebang , Class. Quant. Grav. 40, 015004 (2023), [arXiv:2203.05589 [hep-th]]

    2023 arXiv

  24. [32]

    S. W. Wei, Y. X. Liu, and R. B. Mann, Black Hole Solutions as Topological Thermodynamic Defects , Phys. Rev. Lett. 129, 191101 (2022), [arXiv:2208.01932 [gr-qc]]

    2022 arXiv

  25. [33]

    Wu, Topological classes of rotating black holes , Phys

    D. Wu, Topological classes of rotating black holes , Phys. Rev. D 107, 024024 (2023), [arXiv:2211.15151 [gr-qc]]

    2023 arXiv

  26. [34]

    Wu and S

    D. Wu and S. Q. Wu, Topological classes of thermodynamics of rotating AdS black holes , Phys. Rev. D 107, 084002 (2023), [arXiv:2301.03002 [hep-th]]

    2023 arXiv

  27. [35]

    Shao-Wen Wei, Yu-Xiao Liu, and Robert B. Mann. Universal topological classifications of black hole thermodynamics , Phys. Rev. D 110, L081501 (2024), [arXiv:2409.09333 [gr-qc]]

    2024 arXiv

  28. [36]

    M. J. Duff and J. T. Liu, Anti-de Sitter black holes in gauged N = 8 supergravity , Nucl. Phys. B 554, 237 (1999), [arXiv:hep-th/9901149 [hep-th]]

    1999 arXiv

  29. [37]

    Behrndt, M

    K. Behrndt, M. Cvetic, and W. A. Sabra, Nonextreme black holes of five-dimensional N=2 AdS supergravity , Nucl. Phys. B 553, 317 (1999), [arXiv:hep-th/9810227 [hep-th]]

    1999 arXiv

  30. [38]

    D. Wu, S. Y. Gu, X. D. Zhu, Q. Q. Jiang, and S. Z. Yang, Topological classes of thermodynamics of the static multi-charge AdS black holes in gauged supergravities: novel temperature-dependent thermodynamic topological phase transition , JHEP 06, 213 (2024), [arXiv:2402.00106 [hep-th]]

    2024 arXiv

  31. [39]

    H. S. Liu, Z. F. Mai, Y. Z. Li, and H. L\"u, Quasi-topological Electromagnetism: Dark Energy, Dyonic Black Holes, Stable Photon Spheres and Hidden Electromagnetic Duality , Sci. China Phys. Mech. Astron. 63, 240411 (2020), [arXiv:1907.10876 [hep-th]]

    2020 arXiv

  32. [40]

    Z. Q. Chen and S. W. Wei, Thermodynamical topology with multiple defect curves for dyonic AdS black holes , Eur. Phys. J. C 84, 1294 (2024), [arXiv:2405.07525 [hep-th]]

    2024 arXiv

  33. [41]

    D. Wu, W. Liu, S. Q. Wu, and R. B. Mann, Novel topological classes in black hole thermodynamics , Phys. Rev. D 111, L061501 (2025), [arXiv:2411.10102 [hep-th]]

    2025 arXiv

  34. [42]

    S. W. Wei and Y. X. Liu, Topology of black hole thermodynamics , Phys. Rev. D 105, 104003 (2022), [arXiv:2112.01706 [gr-qc]]

    2022 arXiv

  35. [43]

    P. C. W. Davies, Thermodynamics of Black Holes , Proc. Roy. Soc. Lond. A 353, 499 (1977)

    1977

  36. [44]

    P. C. W. Davies, Thermodynamic Phase Transitions of Kerr-Newman Black Holes in De Sitter Space , Class. Quant. Grav. 6, 1909 (1989)

    1989

  37. [45]

    C. O. Lousto, The Fourth law of black hole thermodynamics , Nucl. Phys. B 410, 155 (1993), [erratum: Nucl. Phys. B 449, 433 (1995)], [arXiv:gr-qc/9306014 [gr-qc]]

    1993 arXiv

  38. [46]

    C. O. Lousto, The Emergence of an effective two-dimensional quantum description from the study of critical phenomena in black holes , Phys. Rev. D 51, 1733 (1995), [arXiv:gr-qc/9405048 [gr-qc]]

    1995 arXiv

  39. [47]

    J. P. Muniain and D. D. Piriz, Critical behavior of dimensionally continued black holes , Phys. Rev. D 53, 816 (1996), [arXiv:gr-qc/9502029 [gr-qc]]

    1996 arXiv

  40. [48]

    Bhattacharya, K

    K. Bhattacharya, K. Bamba, and D. Singleton, Topological interpretation of extremal and Davies-type phase transitions of black holes , Phys. Lett. B 854, 138722 (2024), [arXiv:2402.18791 [gr-qc]]

    2024 arXiv

  41. [49]

    Z. Y. Fan, Topological interpretation for phase transitions of black holes , Phys. Rev. D 107, 044026 (2023), [arXiv:2211.12957 [gr-qc]]

    2023 arXiv

  42. [50]

    P. K. Yerra, C. Bhamidipati, and S. Mukherji, Topology of critical points and Hawking-Page transition , Phys. Rev. D 106, 064059 (2022), [arXiv:2208.06388 [hep-th]]

    2022 arXiv

  43. [51]

    C. Fang, J. Jiang, and M. Zhang, Revisiting thermodynamic topologies of black holes , JHEP 01, 102 (2023), [arXiv:2211.15534 [gr-qc]]

    2023 arXiv

  44. [52]

    Y. S. Duan and M. L. Ge, SU(2) Gauge Theory and Electrodynamics with N Magnetic Monopoles , Sci. Sin. 9, 1072 (1979)

    1979

  45. [53]

    Y. S. Duan, The structure of the topological current , Stanford Linear Accelerator Center (SLAC), Report No. SLAC-PUB-3301, 1984

    1984

  46. [54]

    J. W. York, Jr., Black hole thermodynamics and the Euclidean Einstein action , Phys. Rev. D 33, 2092 (1986)

    1986

  47. [55]

    B. F. Whiting, Black Holes and Thermodynamics , Class. Quant. Grav. 7, 15 (1990)

    1990

  48. [56]

    Kastor, S

    D. Kastor, S. Ray, and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes , Class. Quant. Grav. 26, 195011 (2009), [arXiv:0904.2765 [hep-th]]

    2009 arXiv

  49. [57]

    B. P. Dolan, Pressure and volume in the first law of black hole thermodynamics , Class. Quant. Grav. 28, 235017 (2011), [arXiv:1106.6260 [gr-qc]]

    2011 arXiv

  50. [58]

    B. P. Dolan, A. Kostouki, D. Kubiznak, and R. B. Mann, Isolated critical point from Lovelock gravity , Class. Quant. Grav. 31, 242001 (2014), [arXiv:1407.4783 [hep-th]]

    2014 arXiv

  51. [59]

    A. M. Frassino, R. B. Mann, and F. Simovic, Critical points in Lovelock Black Holes , [arXiv:1611.03525 [hep-th]]

    arXiv

  52. [60]

    M. B. Ahmed, D. Kubiznak, and R. B. Mann, Vortex-antivortex pair creation in black hole thermodynamics , Phys. Rev. D 107, 046013 (2023), [arXiv:2207.02147 [hep-th]]

    2023 arXiv

  53. [61]

    R. G. Cai, L. M. Cao, and N. Ohta, Black Holes in Gravity with Conformal Anomaly and Logarithmic Term in Black Hole Entropy , JHEP 04, 082 (2010), [arXiv:0911.4379 [hep-th]]

    2010 arXiv

  54. [62]

    R. G. Cai, Thermodynamics of Conformal Anomaly Corrected Black Holes in AdS Space , Phys. Lett. B 733, 183 (2014), [arXiv:1405.1246 [hep-th]]

    2014 arXiv

  55. [63]

    Y. P. Hu, Y. S. An, G. Y. Sun, W. L. You, D. N. Shi, H. Zhang, X. Chen, and R. G. Cai, Quantum anomaly triggers the violation of scaling laws in gravitational system , [arXiv:2410.23783 [gr-qc]]

    arXiv

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Reviewed August 6, 2026 · model on record in the stance chip above.